Section: Research Program

Stochastic and agent-based models

The link between stochastic processes and kinetic equations is a domain already present in our research  (H. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: a comparison, J. Math. Biol, 2009)  [67] and that we plan to develop further. They can be viewed either as complementary approaches, useful to take into account different scales (smaller scales for stochastic models, larger scales for mean-field limits), or even as two different viewpoints on the same problem  [66], enriching each other. Neuroscience is a domain where this is particularly true because noise contributes significantly to the activity of neurons; this is the case of networks where mean field limits are derived from stochastic individual-based models and lead to fundamental questions on the well-posedness and behaviours of the system  (Cáceres, Carrillo, Perthame J. Math. Neurosci. 2011; Pakdaman, Perthame, Salort Nonlinearity 2010). One strength and originality of our project is our close connection and collaboration not only with probability theorists but also with statisticians, who provide us with efficient help in the identification of our model parameters.

Agent-based systems consider each component individually. For example, in multi-cellular system modelling, the basic unit is the cell, and each cell is considered  [70], [89]. This approach has advantages if the population of cells reveals heterogeneities on small spatial scales as it occurs if organ architecture is represented  [76], or if the number of cells in a particular state is small. Different approaches have been used to model cellular agents in multi-cellular systems in space, roughly divided in lattice models (e.g.  [85]) and in lattice-free (or off-lattice) models, in which the position  [70], [73] or even the shape (e.g.  [89]) of the cell can change gradually.

The dynamics of cells in lattice-based models is usually described by rules chosen to mimic the behaviour of a cell including its physical behavior. The advantage of this approach is that it is simpler and that simulation times for a given number of cells are shorter than in lattice-free models. In contrast, most lattice-free models attempt to parameterise cells by measurable values with a direct physical or biological meaning, hence allowing identification of physiologically meaningful parameter ranges. This improves model simulation feasibility, since parameter sensitivity analyses in simulations shows significant improvements when a high dimensional parameter space can be reduced. It also facilitates the development of systematic systems biology and systems medicine strategies to identify mechanisms underlying complex tissue organisation processes ( [89], [71]).

Moreover, it is straightforward to include relevant signal transduction and metabolic pathways in each cell within the framework of agent-based models, which is a key advantage in the present times, as the interplay of components at many levels is more and more precisely studied  [91].